Extreme k-families

Let P be a poset. A subset A of P is a k-family ii A contains no (k+1)-element chain. For i 1, let A i be the set of elements of A at depth i?1 in A. The k-families of P can be ordered by deening A B ii for all i, A i is included in the order ideal generated by B i. This paper examines minimal r-element k-families, deened as k-families A such that A = r and for every B < A, B < r. Minimal k-families are related to maximal r-antichains and an operation called Sperner closure which have been used to obtain extremal results for families of sets with width restrictions. Let M k;r be the set of minimal r-element k-families and let M k = S r0 M k;r. It is shown that M k is a join-subsemilattice of the lattice A k of k-families. M k is a lower semimodular lattice where the r'th rank is given by M k;r. If w k is the 1 maximum size of a k-family, then M k;r ? wk r and S M k P wk i=1 d i k e. Let D(A) = maxf B ? A j B is a k-family and B Ag. For k-families A and B, D(A _ B) D(A) + D(B). This result shows that fA j D(A) = 0g is also a join-subsemilattice of A k .